38 Chapter 2. The AdS/CFT correspondence
is the imaginary-time formalism which basically Euclideanizes the time-coordinatetby Wick rotatationt → −iτEuclid and afterwards compactifies it on a circle with periodβ = 1/T such thatτ +β ∼ τ. Any correlation function defined on this periodic Euclidean space-time can be Fourier-transformed to the four momentum coordinates~k. Because of the periodicity and limited range in the time-coordinate0 ≤ τ ≤ β the Fourier frequency k0 is discrete k0 = 2πT n, n = 0, 1, . . .. These are the real-valued Matsubara frequencies. The disadvantage here is that we basically trade the time coordinate for temperature and therefore loose any notion of temporal evolution of our system. Therefore we can only describe equilibrium states with this formalism. In order to incorporate time and temperature at equal footing we need to employ the more complicated real-time formalism. We will come back to this issue when discussing correlation functions in section 3.1.
If we have the notion of a temperature in our quantum field theory, we can also define a chemical potential µ for a conserved total charge Q = R
volumeJ0 with a charge density J0. Here we assume that the chemical potentialµis constant with respect to the four Minkowski directions~x. The chemical potential is a measure for the energy needed to add one unit of chargeQto the thermal system and it is given in terms of the grandcanonical potential in the grandcanonical ensemble as
µ=−∂J0Ω. (2.83)
In order to prove this recall also that a system in contact only with a heat bath is described by the canonical ensemble with the partition function
Zcanonical =e−βRH, (2.84)
with the Hamiltonian density H giving the energy of the system after integrating over the volume. If we would like to work at a finite chemical potential, in addition we need to put our system into contact with a particle bath. Then the relevant ensemble is the grandcanonical one with the partition function
Zgrand =e−βR(H−µJ0). (2.85)
The finite charge density J0 is the thermodynamically conjugate variable to the chemical potential. Introducing a finite charge density will also change the chemical potential while changing the chemical potential will in general also change the charge density. In the grand canonical ensemble the grandcanonical potential is defined by
Ω = −1
β lnZgrand = Z
(H −µJ0), (2.86)
which immediately confirms the chemical potential formula (2.83).
Now a chemical potential in a thermal QFT is given by the time component of a gauge field A0. This may be seen heuristically by comparing the partition function in the grand canonical ensemble (including the charge densityJ0) on one hand
Z =e−βR(H−µJ0) (2.87)
with the partition function at zero charge density but for a gauge theory including a gauge fieldAµcoupling to the conserved currentJµon the other hand
Z[Aµ] =e−βR(H−AµJµ). (2.88)
2.4. AdS/CFT at finite temperature 39
Choosing only the time component of the gauge field Aµ non-zero and having called the thermodynamical charge density suggestivelyJ0, we can now identify
A0 =µ . (2.89)
Thus we have seen that introducing a finite gauge field time component in a thermal QFT is equivalent to (and therefore may be interpreted as) the introduction of a finite chemical poten-tialµfor the charge densityJ0. A more formal treatment of this may be found in section 3.2.2.
Introducing temperature In order to study thermal gauge theories through AdS/CFT we need a notion of temperature on the gravity side. This means that we need to modify the background and in particular the background metric in order to incorporate temperature in the dual gauge theory. The idea of using a metric describing the geometry of a black hole comes about quite naturally since black holes are holographic thermal objects themselves whose d-dimensional exterior physics is completely captured by their (d− 1)-dimensional horizon surface. This phenomenon is studied in the field called black hole thermodynamics.
The Bekenstein-Hawking formula relates the area of the black hole horizon to the entropy of the complete black hole (bulk) which has a distinct Hawking temperature depending on its mass.
It was first proposed in [84] that black hole backgrounds or black branes as described in section 2.1.2 are holographically dual to a gauge theory at finite temperature. The metric for a stack of black D3-branes can be conveniently written in the form
ds2 = 1 2
̺ L
2
−f2
f˜dt2+ ˜fd~x2
+ L
̺ 2
d̺2+̺2dΩ52
, (2.90)
with
f(̺) = 1− r04
̺4 , f(̺) = 1 +˜ r04
̺4 . (2.91)
We obtain this form of the metric from (2.18) by the transformation ̺2 = r2 +√
r4−r04
wherer0 is the location of the horizon. The Hawking temperatureTH of the black hole hori-zon is equivalent to the temperature T in the thermal gauge theory on the other side of the correspondence. In order to relate the temperature T to the factors appearing in metric com-ponents, we make the metric Euclidean by Wick rotation. Demanding regularity at the horizon renders the Euclidean time coordinate to be periodic with periodβ = 1/T =r0/(πL2). Note, that this background is confining [84] and preserves all the supersymmetry, i.e. the dual field theory is N = 4SYM at finite temperature. Further there exist crucial differences between the Euclideanized background and its Minkowski version. We will discuss this issue in sec-tion 3.1.1.
Quarks & chemical potential In order to include fundamental matter in this finite tem-perature setup we introduce D7-probe branes as described in section 2.3. At vanishing baryon density it was observed in [37] that these thermal D7-embeddings are special because in the gauge theory a phase transition appears which is dual to a geometric transition on the gravity
40 Chapter 2. The AdS/CFT correspondence
Figure 2.3: Increasing the temperature from the left to the right picture we see that the black hole becomes larger. The embedded brane is pulled towards the horizon stronger and stronger until the probe brane just touches the black hole horizon (middle picture). Increasing the temperature further the brane is pulled through the horizon.
This picture is taken from [56].
side (see figure 2.3). The setup is governed by a parameter m ∝ Mq/T which is propor-tional to the quotient of quark mass Mq and temperature T. At large values of m we have Minkowski embeddings which end outside the horizon. We write down the black hole metric in the coordinates introduced in (2.72)
ds2 =
˜
w2+wH4
˜ w2
dx2+ ( ˜w4−wH4)2
˜
w2( ˜w4−wH4)dt2+ 1 + (∂̺w6)2
˜
w2 d̺2+ ̺2
˜
w2dΩ32, (2.92) where we definew˜2 =̺2+w6(̺)2 andwH is the location of the horizon. In theAdS5×S5 -background the D7-brane fills the AdS wrapping anS3inside theS5. Looking at theS3-part of the metric (2.92), forρ= 0, , w6 > wHwe find that theS3shrinks to zero size before reaching the horizon. These Minkowski embeddings resemble those present at vanishing temperature at large values ofm.
Decreasing the parameterm we reach a critical value below which the D7-brane always reaches to the horizon. The geometrical difference is that for these black hole embeddings now theS1in time direction collapses as can be seen from the time component of the metric (2.92).
This means that the D3/D7-system in presence of a black hole undergoes a geometrical transition. That transition is dual to a first order phase transition in the thermal field theory dual. The physics of this transition is discussed in greater detail in section 4.3.
However, the central achievement of this present work is to introduce a finite baryon and isospin density in the setup we have just described. We will see that this changes the em-beddings and also the phase structure of the theory. We will further observe that the phase transition is softened. This statement will be explained in the discussion of this system’s hydrodynamics in chapter 6. We discover a further transition at equal baryon and isospin densities discussed in the thermodynamics section 4.4.
Brane thermodynamics and holographic renormalization At finite temperature an ex-tension of the standard AdS/CFT claim is the conjecture that the thermodynamics of the